Tag: Hooke’s Law

Recently I’ve been going back and forth with various folks in the twitter realm about the fundamentals of physics. The biggest stumbling block is the issue of forces and how to define them. I stated in a previous post that you need 3 specific attributes to calculate any force equation:

Magnitude

Vector (or Direction)

Time

I then present a simple example of a 4000kg car accelerating at 10 m/s² for 10s. The Impulse of Force of the car at each second is:

Mass x acceleration x time¹ = Impulse of Force¹ (¹ is all reference frames from 1 – 10 seconds). Multiplying Force · Time is known as impulse and is a valid way of expressing a force.

However, where I caught the consternation of others in the twitter realm was due to confounding of terms that are very specific in the physics world. We all experience the affects of forces all day, everyday in our lives and we generally don’t break them down into units of measurement, types of units, work vs. force, etc. So my way of describing what I see as a fatal flaw in gravity was met with irate rantings.

For one, I was pointing out a potential problem with something they hold in an almost sacred sense (gravity). And secondly, I was not using terminology correctly for their taste. I don’t necessarily blame them for wanting exact language, but I would argue that they most likely understood the point I was trying to make but did not want point me in the right direction. But, for the most part however, they were decent. In a way, by them resisting what I was presenting forced me to fix and clarify my position.

Force, rate of change, work and energy

What I saw was that over time the acceleration due to gravity should equate to an increase in force on the mass involved. However, the concept of force is important since in the case of gravity it is, indeed, constant. I wasn’t arguing that gravity is increasing but the net forces are increasing. I kept arguing that time must be included in the equation or it doesn’t represent reality (though it was valid see impulse of the force). But again, my use of the words net force with time was incorrect with respect to physics as we are taught today and net forces has a specific meaning. It wasn’t until I recalled kinetic and potential energy (work) and how it relates back to force that I was able to present my case in a language that would be acceptable.

From the car example above, if F = ma, then the rate of change is 40,000 N which means the force is constant. But this is counter-intuitive to most people since they know the car is traveling faster with each second. However, for acceleration to exist at all, energy must be constantly applied. If we accept the conservation of energy law, then that energy must be going somewhere. In this case it is being expressed as kinetic energy.

In the case of gravity acting on an object or a person on the earth’s surface, gravity is that constant force or energy that is supplying the acceleration. Either the object or person needs to move or the energy needs to be converted into something else like heat or sound or whatever. It is the amount of energy that is increasing not the force itself. We can conclude then, that time multiplies how much energy is in a system with respect to the force being applied.

We can all bear witness to the fact that we are not heating up or emitting noise or expressing some other form of energy release under the stress if gravity. The counter argument is that since there is no movement there can be no acceleration. But this is a fallacy since the height of the person is equal to the “x” value in Hooke’s law and the motion is expressed as the compression of the human frame.

There is also a distinction between kinetic energy and potential energy. A car driving at 100 km/h has a specific amount of kinetic energy but a spring under increasing compression has potential energy that is being stored in the spring itself. The spring has a maximum compression it can reach before it will begin to become crushed under an increasing load.

Hooke’s law is only a first-order linear approximation to the real response of springs and other elastic bodies to applied forces. It must eventually fail once the forces exceed some limit, since no material can be compressed beyond a certain minimum size, or stretched beyond a maximum size, without some permanent deformation or change of state. Many materials will noticeably deviate from Hooke’s law well before those elastic limits are reached.

Even though the spring no longer has motion the amount energy will increase as the load capacity is transferred to less elastic structures like the metal of the spring or the surface the spring is sitting on.

In other words, the increase of potential energy will propagate to the surrounding environment and begin to compress the weakest structures first. In the case of a human being standing on the surface of the earth, the acceleration due to gravity would continue to propagate through the human frame as per Hooke’s law. I have had many discussions with folks who insist that the forces between the earth and the human being cancel each other out. But they are confusing force with work.

Any equal and opposite reaction is within the frame of the human being (like a spring) and the ground they are standing on. The opposing forces balance out, but this would not stop the acceleration; the balanced forces would only resist the downward pull of gravity and subsequently increase the potential energy. Since gravity is supposed to be an acceleration and is supposed to be pulling at a force proportional to the mass of the earth, the human being wouldn’t stand a chance. It is precisely because of the equal and opposite reaction and the balancing of forces that potential energy is possible. So they are confusing the forces involved with the energy (work) being applied to the mass.

Potential Energy

assuming conservation of energy: if F=ma and a = 9.8 m/s² and m > 0 then ΔPE > 0. where’s all that energy going? We should all be crushed by now.

Spring energy

The potential energy Uel(x) stored in a spring is given by

which comes from adding up the energy it takes to incrementally compress the spring. That is, the integral of force over displacement. Since the external force has the same general direction as the displacement, the potential energy of a spring is always non-negative.

This potential Uel can be visualized as a parabola on the Ux-plane such that Uel(x) = 1/2kx2. As the spring is stretched in the positive x-direction, the potential energy increases parabolically (the same thing happens as the spring is compressed). Since the change in potential energy changes at a constant rate:

Note that the change in the change in U is constant even when the displacement and acceleration are zero.

What does this mean? It means, a constant and catastrophic amount of potential energy should be building up in every object on the earth’s surface. As mentioned above, this does not violate F=ma since we are not talking about an increase in the force of gravity but an increase in potential energy due to that constant force.

If we calculate the “elastic” capacity of the human frame and equate it to “k” and multiplied that by “x” which would be the height of the person (since they should be getting compressed by gravity) and then multiply by 1/2 we should get the potential energy stored or expressed within the human frame. Also, since the value for “k” of the human being is less than “k” for ground, the human being would be crushed into the ground. However, as we all know, we aren’t springs, and won’t bounce back from such an event. Most of the energy will be released in the form of noise and heat. Not a pretty picture.

The modern theory of elasticity generalizes Hooke’s law to say that the strain (deformation) of an elastic object or material is proportional to the stress applied to it. However, since general stresses and strains may have multiple independent components, the “proportionality factor” may no longer be just a single real number, but rather a linear map (a tensor) that can be represented by a matrix of real numbers.